X iv : h ep - p h / 99 07 34 0 v 2 1 5 Ju l 1 99 9 1 What QCD Tells Us About Nature – and Why We Should

1.1. It embodies deep and beautiful principles. These are, first of all, the general principles of quantum mechanics, special relativity, and locality, that lead one to relativistic quantum field theory [1]. In addition, we require invariance under the nonabelian gauge symmetry SU(3), the specific matter content of quarks – six spin 1/2 Dirac fermions which are color triplets – and renormalizability. These requirements determine the theory completely, up to a very small number of continuous parameters as discussed below. Deeper consideration reduces the axioms further. Theoretical physicists have learned the hard way that consistent, non-trivial relativistic quantum field theories are difficult to construct, due to the infinite number of degrees of freedom (per unit volume) needed to construct local fields, which tends to bring in ultraviolet divergences. To construct a relativistic quantum theory, one typically introduces at intermediate stages a cutoff, which spoils the locality or relativistic invariance of the theory. Then one attempts to remove the cutoff, while adjusting the defining parameters, to achieve a finite, cutoff-independent limiting theory. Renormalizable theories are those for which this can be done, order by order in a perturbation expansion around free field theory. This formulation, while convenient for mathematical analysis, obviously begs the question whether the perturbation theory converges (and in practice it never does). A more straightforward procedure, conceptually, is to regulate the theory as a whole by discretizing it, approximating space-time by a lattice [2]. This spoils the continuous space-time symmetries of the theory. Then one attempts to remove dependence on the discretization by refining it, while if necessary adjusting the defining parameters, to achieve a finite limiting theory that does not depend on the discretization, and does respect the space-time symmetries. The redefinition of parameters is necessary, because in refining